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Math: Probability

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Re: Math: Probability [#permalink]

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New post 19 Jul 2017, 20:19
This is very basic but I am getting confused by the concepts.

If a fair coin is flipped three times, what is the probability that it
comes up heads all three times?

So the way I understand it, P of getting tails is 1/2*1/2*1/2=1/8

Because each event is independent from one another, I calculate the probability of getting heads as 1-1/8=7/8 but the right answer is 1/8....

What am I doing wrong?

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Re: Math: Probability [#permalink]

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New post 19 Jul 2017, 22:07
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azelastine wrote:
This is very basic but I am getting confused by the concepts.

If a fair coin is flipped three times, what is the probability that it
comes up heads all three times?

So the way I understand it, P of getting tails is 1/2*1/2*1/2=1/8

Because each event is independent from one another, I calculate the probability of getting heads as 1-1/8=7/8 but the right answer is 1/8....

What am I doing wrong?


Hi azelastine,

Your first part is correct, i.e. probability of getting 3 tails is 1/8.
Similarly, you can compute the probability of getting three heads.
Please note that each coin toss is independent of one another, that's why we can multiply the probability of each event.
P(HHH) = P(getting head on the first toss) and P(getting head on the second toss) and P(getting head on the third toss)= 1/2 * 1/2 * 1/2 = 1/8

AND implies Multiplication(*)
OR implies Addition(+)

Quote:
Because each event is independent from one another, I calculate the probability of getting heads as 1-1/8=7/8 but the right answer is 1/8....


Here you are assuming that following events are complementary to each other:
1. Getting three tails, i.e. P(TTT) and
2. Getting three heads, i.e.P(HHH)
P(TTT) + P(HHH) = 1
This is not correct.

Let's enumerate the all possible outcomes(sample space).
1. HHH
2. HHT
3. HTT
4. TTT
5. TTH
6. THH
7. THT
8. HTH

Sum of probabilities of all these events will be equal to 1. i.e.
P(HHH) + P(HHT) + P(HTT) + P(TTT) + P(TTH) + P(THH) + P(THT) + P(HTH) = 1

In general, if you toss the coin n times, then total possible outcomes = 2^n. In this cane, n= 3, so total outcomes = 2^3 = 8.

Hope it helps.

Thanks.

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Re: Math: Probability [#permalink]

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New post 22 Jul 2017, 12:51
ganand wrote:
azelastine wrote:
This is very basic but I am getting confused by the concepts.

If a fair coin is flipped three times, what is the probability that it
comes up heads all three times?

So the way I understand it, P of getting tails is 1/2*1/2*1/2=1/8

Because each event is independent from one another, I calculate the probability of getting heads as 1-1/8=7/8 but the right answer is 1/8....

What am I doing wrong?


Hi azelastine,

Your first part is correct, i.e. probability of getting 3 tails is 1/8.
Similarly, you can compute the probability of getting three heads.
Please note that each coin toss is independent of one another, that's why we can multiply the probability of each event.
P(HHH) = P(getting head on the first toss) and P(getting head on the second toss) and P(getting head on the third toss)= 1/2 * 1/2 * 1/2 = 1/8

AND implies Multiplication(*)
OR implies Addition(+)

Quote:
Because each event is independent from one another, I calculate the probability of getting heads as 1-1/8=7/8 but the right answer is 1/8....


Here you are assuming that following events are complementary to each other:
1. Getting three tails, i.e. P(TTT) and
2. Getting three heads, i.e.P(HHH)
P(TTT) + P(HHH) = 1
This is not correct.

Let's enumerate the all possible outcomes(sample space).
1. HHH
2. HHT
3. HTT
4. TTT
5. TTH
6. THH
7. THT
8. HTH

Sum of probabilities of all these events will be equal to 1. i.e.
P(HHH) + P(HHT) + P(HTT) + P(TTT) + P(TTH) + P(THH) + P(THT) + P(HTH) = 1

In general, if you toss the coin n times, then total possible outcomes = 2^n. In this cane, n= 3, so total outcomes = 2^3 = 8.

Hope it helps.

Thanks.


Thanks for the explanation, this is very helpful.

Could you help me understand the difference between this and the following problem?

A fair coin is flipped twice. What is the probability that the coin lands showing heads on the first flip, second flip or both?

The solution to this question is two-step: 1) P (two tails)=1/4, 2) 1-1/4=3/4

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Re: Math: Probability [#permalink]

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New post 28 Jul 2017, 04:30
azelastine wrote:
Thanks for the explanation, this is very helpful.

Could you help me understand the difference between this and the following problem?

A fair coin is flipped twice. What is the probability that the coin lands showing heads on the first flip, second flip or both?

The solution to this question is two-step: 1) P (two tails)=1/4, 2) 1-1/4=3/4


Hi azelastine,

I'll try to explain the difference.

Quote:
A fair coin is flipped twice. What is the probability that the coin lands showing heads on the first flip, second flip or both?


Sample space: {HH, HT, TH, TT}

Favorable event = {HH, HT, TH} and complementary event = {TT}

In two ways you can solve this question.

1. Compute the probability of the favorable event
In this case required probability = P(HH) + P(HT) + P(TH) =\(\frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2} = \frac{3}{4}\)
or
2. Compute the probability of complementary event and subtract it from 1. You have solved it using this method.

In general, when a complementary event is small then use the 2nd approach.
The complementary approach can also be used in P&C questions.

Please go through the first post of the thread. These points are discussed in detail with examples.

Hope this helps.

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Re: Math: Probability [#permalink]

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New post 12 Sep 2017, 02:06
Q: If the probability of a certain event is p, what is the probability of it occurring k times in n-time sequence?
(Or in English, what is the probability of getting 3 heads while tossing a coin 8 times?)
Solution: All events are independent. So, we can say that:

P′=pk∗(1−p)n−kP′=pk∗(1−p)n−k (1)

But it isn't the right answer. It would be right if we specified exactly each position for events in the sequence. So, we need to take into account that there are more than one outcomes. Let's consider our example with a coin where "H" stands for Heads and "T" stands for Tails:
HHHTTTTT and HHTTTTTH are different mutually exclusive outcomes but they both have 3 heads and 5 tails. Therefore, we need to include all combinations of heads and tails. In our general question, probability of occurring event k times in n-time sequence could be expressed as:

P=Cnk∗pk∗(1−p)n−kP=Ckn∗pk∗(1−p)n−k (2)

In the example with a coin, right answer is P=C83∗0.53∗0.55=C83∗0.58

Aren't we looking for the permutations here instead of the combination ? aren't HHHTTTTT and HHTTTTTH the same combination but two different permutations?
Thank you for your clarifications ;)

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Re: Math: Probability [#permalink]

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New post 21 Sep 2017, 20:52
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Quote:
Aren't we looking for the permutations here instead of the combination? aren't HHHTTTTT and HHTTTTTH the same combination but two different permutations?


Hi Aminaelm ,

Yes, you are right. Let's analyze it more closely.

No. of arrangements of HHHTTTTT = \(\frac{8!}{5!\times 3!} = {{8}\choose{3}}\) .

Hence, we are taking into account all the possible arrangements in the given formula. Hope this helps.

Thanks.

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Re: Math: Probability   [#permalink] 21 Sep 2017, 20:52

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